Question
Solve the following quadratic equations by factorization:$x^2 - x - a(a + 1) = 0$
✓
Answer
$x^2 - x - a(a + 1) = 0$
$\Rightarrow x^2 + {(a) - (a + 1)}x - a(a + 1) = 0$
${ 1 = a - a + 1}$
$\Rightarrow x^2 + ax - (a + 1)x - a(a + 1) = 0$
$\Rightarrow x(x + a) - (a + 1)(x + a) = 0$
$\Rightarrow (x + a)(x - a - 1) = 0$
Either $x + a = 0$, then $x = -a$
or $x - a - 1 = 0$, then $x = a + 1$
$\therefore$ Roots are $-a, a + 1$
$\Rightarrow x^2 + {(a) - (a + 1)}x - a(a + 1) = 0$
${ 1 = a - a + 1}$
$\Rightarrow x^2 + ax - (a + 1)x - a(a + 1) = 0$
$\Rightarrow x(x + a) - (a + 1)(x + a) = 0$
$\Rightarrow (x + a)(x - a - 1) = 0$
Either $x + a = 0$, then $x = -a$
or $x - a - 1 = 0$, then $x = a + 1$
$\therefore$ Roots are $-a, a + 1$
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